## Pumping Lemma

Show that the set $\displaystyle \{0^{2n}1^{n}\}$ is not regular using the pumping lemma.

$\displaystyle \mbox{Pumping lemma: if }M=(S,I,f,s_{0},F)$ $\displaystyle \mbox{ is a deterministic finite automaton and if } x \mbox{ is a string in }L(m),$$\displaystyle \mbox{ the language recognized by M, with }$$\displaystyle l(x) \geq |S|,$ $\displaystyle \mbox{ then there are strings }u, v, w \mbox{ in } I^{*}$ $\displaystyle \mbox{such that }$ $\displaystyle x = uvw, I(uv) \leq |S| \mbox{ and }l(v) \geq 1,\mbox{ and }uv^{i}w \in L(M) \mbox{ for }i = 0, 1, 2, ...$